Elastic PMLs for Resonator Anchor Loss Simulation

59 Pages·2005·1 MB·English

Elastic PMLs
for
Resonator Anchor Loss Simulation
1 2
David Bindel and Sanjay Govindjee
1
Department of Electrical Engineering and Computer Science
2
Department of Civil Engineering
University of California at Berkeley
D.Bindel,USNCCM8–p.1/45
Contributors
Tsuyoshi Koyama – PhD Student, Civil Engineering
Wei He – PhD Student, Civil Engineering
Emmanuel Quévy – Postdoc, Electrical Engineering
Roger Howe – Professor, Electrical Engineering
James Demmel – Professor, Computer Science
D.Bindel,USNCCM8–p.2/45
Outline
Electromechanical resonators and RF MEMS
Damping and quality of resonance
Anchor losses and Perfectly Matched Layers
Analysis of the discretized PMLs
Complex symmetry and structured model reduction
Analysis of a disk resonator
Conclusions
D.Bindel,USNCCM8–p.3/45
Outline
Electromechanical resonators and RF MEMS
Damping and quality of resonance
Anchor losses and Perfectly Matched Layers
Analysis of the discretized PMLs
Complex symmetry and structured model reduction
Analysis of a disk resonator
Conclusions
D.Bindel,USNCCM8–p.4/45
How many MEMS?
D.Bindel,USNCCM8–p.5/45
Why resonant MEMS?
Microguitars from Cornell University (1997 and 2003)
Sensing elements (inertial, chemical)
Frequency references
Filter elements
Neural networks
Really high-pitch guitars
D.Bindel,USNCCM8–p.6/45
Micromechanical filters
Mechanical filter
Radio signal
Filtered signal
Capacitive drive Capacitive sense
Mechanical high-frequency (high MHz-GHz) filter
Your cell phone is mechanical!
Advantage over quartz surface acoustic wave filters
Integrated into chip
Low power
Success =⇒ “Calling Dick Tracy!”
D.Bindel,USNCCM8–p.7/45
Designing transfer functions
Time domain:
00 0
Mu + Cu + Ku = bφ(t)
T
y(t) = p u
Frequency domain:
2 ˆ
−ω Muˆ + iωCuˆ + Kuˆ = bφ(ω)
T
yˆ(ω) = p u
Transfer function:
T 2 −1
H(ω) = p (−ω M + iωC + K) b
ˆ
yˆ(ω) = H(ω)φ(ω)
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Damping and filters
20 log |H(ω)|
10
ω
Want “sharp” poles for narrowband filters
=⇒ Want to minimize damping
Electronic filters have too much
Understanding of damping in MEMS is lacking
D.Bindel,USNCCM8–p.9/45
Damping and Q
Designers want high quality of resonance (Q)
Dimensionless damping in a one-dof system:
2
d u du
−1
+ Q + u = F (t)
dt2 dt
C
For a resonant mode with frequency ω ∈ :
|ω| Stored energy
Q := =
2 Im(ω) Energy loss per radian
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